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Full text of "Analytical Mechanics"

256                       ANALYTICAL MECHANICS
where r is the density of water.   Substituting this expression for m in equation (1)
0 (a + )=! [(a + )] at
= (a + kt)*^ + Z(a-rkt)*kv at
,                                  dv ,     3k           *                                   ,~
and                                 it + ^+ktv = g>                                    (2>'
r s k                  r 3 k
the integral of which is v = e    a+kt    |  Cge a + kt   dt + c | 
/.    v = e~31 <fl + *3 [JV10* (a + *<> & 4- cl
= (a + ta)~3 [V J(a + &03 d + cl
/    .7^  .T   ( v , 3a2^2 , 3afe2^3 , ^4\   ,   I = (a + ^)-8^ (^ +   + -y- +  J + cj
= (a + ^)-3f| (4 a3^ + 6 a%2 + 4 akH* + kH*) + c] .
Let   # = 0 when t  0;   then   c = 0. fi? 4 a3 + 6 a2Jc^ + 4 a/b2^2 + kH*
PROBLEMS.
1.   Find the pressure upon the canvas roof of a tent produced by a shower.   The following data are given  the raindrops have a velocity of
50  '- at right angles to the roof; the intensity of the shower is such as sec.
to produce a deposit of 0.2 inch per hour; 1 cubic foot of water weighs 62.5 pounds.
2.   Find the pressure on horizontal ground due to the impact of a column of water which falls vertically from a height of 500 feet.
cm
3.   Water flowing through a pipe at the rate of 100  ' is brought to-
sec.
* Equation (2) is of the form ~t- -{- Py = Q, which is the typical linear
equation, with the integral y = e~~*'p<2aT CQe^^dx + cl.